Thesis High-Resolution Photoemission Study of Kondo Insulators ...

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18 Chapter 2. Photoemission Spectroscopy ψk|E0(N) >= 0, Eq. (2.1) is expressed as P < (ɛout) = 1 � δ(ɛout − ɛk)Im (2.2) where τk = � τkiψi and we have used the relation i 2 δ(x) = 1 π Im( 1 x − iη ). Neglecting nondiagonal matrix elements corresponding to different values of i, we can rewrite Eq. (2.2) as P < (ɛout) ∝ � i 1 π � |τki| 2 δ(ɛout − ɛk)ImG < (ɛout − hν), (2.3) k where G < (z) =. (2.4) Here, we further assume that � |τki| 2 δ(ɛout − ɛk) has a negligible energy dependence k within the scanning region on measurement. Then Eq. (2.3) becomes proportional to � ImG < (ɛout − hν), i.e., the momentum-integrated one-particle spectral function i (spectral density of states (DOS)) P < (ɛout) ∝ � i 1 π ImG< (ɛout − hν − iη) (2.5) = A < = (ɛout − hν) (2.6) � | | 2 δ(ɛout − hν − (E0(N) − En(N − 1))).(2.7) kn Without any correlation in electrons, the remaining orbitals are frozen in the final states as they were in the initial state (|En(N − 1) >= ψk|E0(N) >) [2.2] and one can rewrite E0(N) − En(N − 1) as ɛk − W , where ɛk is an energy of the orbital from which a photoelectron is emitted and W is a work function. Then the spectral DOS agrees with the DOS in the ground state as shown in Fig. 2.1: A < (ω) = � 1 π Im 1 � = δ(ω − ɛk). (2.8) ω − ɛk − iη k Here we introduced the energy relative to the Fermi level (EF ) ω = ɛout − hν + W . By substituting ɛ0 k +Σ(k,ω)intoɛk, where ɛ0 k is a bare one-electron energy and Σ(k,ω) is a self energy, we can include the effect of the electron correlation as far as a quasiparticle is well defined. 2 1 η This relation is derived straightforwardly from the representation of δ function δ(x) = π x2 + η2 k
2.1. General Principles 19 Photoelectron Electron Energy DOS Kinetic Energy Photoemission Intensity Fermi level Excitation light Figure 2.1: Principle of valence-band photoemission spectroscopy for the solids at 0 K without any correration in electrons. A photoemission spectrum reproduces the occupied part of DOS. We can also derive the formation for the inverse photoemission spectrum and the energy distribution of the emitted photons is given as P > (E) ∝ � 1 i π ImG> (E − hν − iη) (2.9) = A > (E − hν) (2.10) = � kn | | 2 δ(E − hν − (En(N +1)− E0(N)))(2.11) where E is the energy of the incoming electron and G > 1 (ω) =. (2.12) The photoemission spectrum at finite temperature is straightforwardly given by the Lehmann representation of the temperature (Matsubara) Green function. We consider the grand canonical ensemble and use eigen states |α ′ > and |α ′′ > with their eigen values E ′ and E ′′ instead of |Em(N) > and |En(N − 1) >.
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Page 4 and 5: ii CONTENTS 5.2 Experimental . . .
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Page 18 and 19: 14 References [1.14] F. Iga, M. Kas
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Page 27 and 28: 2.2. Experimental 23 DOS Intensity
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68 Chapter 5. Temperature Dependenc
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74 References [5.10] N. Shimizu, F.
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Chapter 6 Substitution and Temperat
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6.1. Introduction 79 Figure 6.2: Ma
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6.3. Results and Discussion 81 6.3
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6.3. Results and Discussion 83 DOS
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6.3. Results and Discussion 85 dist
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6.3. Results and Discussion 87 Apar
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6.3. Results and Discussion 89 of E
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6.4. Summary 93 and consistent anal
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96 References [6.11] M. J. Rozenber
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98 Chapter 7. Temperature and Co-Su
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100 Chapter 7. Temperature and Co-S
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References [7.1] G. Aeppli and Z. F
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Chapter 8 Temperature and Al-Substi
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8.2. Experimental 113 and Fisk [8.2
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8.3. Results and Discussion 115 8.3
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8.4. Summary 123 temperature magnet
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126 References [8.14] T. Tonogai, A
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128 Chapter 9. Conclusion has shown
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130 Chapter 9. Conclusion (a) FeSi
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132 Chapter 9. Conclusion developed
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